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We consider the one-dimensional Schrödinger equation -f″ + q f = Ef on the positive half-axis with the potential q(r) = (κ - 1/4)r. For each complex number ν, we construct a solution u(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the 'singular' point κ = 0. For -1 < κ < 1 and real ν, the solutions u(E) determine a unitary eigenfunction expansion operator U: L(0,∞) → L(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -∂ + q(r) for the Hamiltonian is diagonalized by the operator U for some ν ∈ R. Using suitable singular Titchmarsh-Weyl m-functions, we explicitly find the measures V and prove their continuity in κ and ν. [ABSTRACT FROM AUTHOR]

We develop a spectrally accurate numerical method to compute solutions of a model PDE used in plasma physics to describe diffusion in velocity space due to Fokker--Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular Sturm--Liouville operator. We present a new algorithm for computing the spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh--Weyl m-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled, and a new formula for the limiting value of the m-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. The complexity of the algorithm is also analyzed, showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired accuracy. A WKB analysis is used to prove that the spectral density function is real analytic. Using this new algorithm, we highlight key properties of the PDE and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations. [ABSTRACT FROM AUTHOR]

We consider the in-plane motion of elastic strings on tree-like network, observed from the ‘leaves’. We investigate the inverse problem of recovering not only the physical properties, i.e. the ‘optical lengths’ of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. To this end we use the Boundary Control method for wave equations on graphs established in [4, 7]. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. [ABSTRACT FROM AUTHOR]

Abstract: Sufficient conditions for the similarity of the operator with an indefinite weight are obtained. These conditions are formulated in terms of Titchmarsh–Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm–Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm–Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case and , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case , we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward–backward” diffusion equations. [Copyright &y& Elsevier]

Asymptotic solutions for the Sturm-Liouville boundary value problem with interior singularities were obtained using asymptotic forms of the Whittaker functions for higher order modes and the Titchmarsh-Weyl m-function for low order modes. However, these split interval techniques do not readily provide the eigenrelations for low order modes. For the first time, with minimal constraints, the eigenvalues for the Sturm-Liouville eigenproblem are obtained when the Titchmarsh-Weyl m-function technique is employed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a certain integral inequality whose associated differential expression contains a periodic potential. In these problems the spectrum consists of bands which may have eigenvalues in the gaps. We use numerical methods to estimate the location of these bands and position of the eigenvalues and use this information to produce new examples of the HELP inequality.